By the union bound, we conclude that $\text{Pr}[v_a\in X]\le1/k$.

Let us now bound the treewidth of $G-X$. For $a\in\{1,\ldots,n\}$, the \emph{$a$-grid} is $F_a=\bigcup_{H\in\HH:i(h)\le a} H$, and we let

the $0$-grid $F_0=\emptyset$. For $a\ge0$, an \emph{$a$-cell} is a maximal connected subset of $\mathbb{R}^d\setminus F_a$.

A set $C\subseteq\mathbb{R}^d$ is a \emph{cell} if it is an $a$-cell for some $a\ge0$.

A cell $C$ is \emph{non-empty} if there exists $v\in V(G-X)$ such that $\iota(v)\subseteq C$.

Note that there exists a rooted tree $T$ whose vertices are

the non-empty cells and such that for $x,y\in V(T)$, we have $x\preceq y$ if and only if $x\subseteq y$.

For each non-empty cell $C$, let us define $\beta(C)$ as the set of vertices $v_i\in V(G-X)$ such that

$\iota(v)\cap C\neq\emptyset$ and $C$ is an $a$-cell for some $a\ge i$.

Let us show that $(T,\beta)$ is a tree decomposition of $G-X$. For each $v_j\in V(G-X)$, the $j$-grid is disjoint from $\omega(v_j)$,

and thus $\iota(v_j)\subseteq\omega(v_j)\subset C$ for some $j$-cell $C\in V(T)$ and $v_j\in\beta(C)$. Consider now an edge $v_iv_j\in E(G-X)$, where $i<j$.

We have $\omega(v_j)\cap\iota(v_i)\neq\emptyset$, and thus $\iota(v_i)\cap C\neq\emptyset$ and $v_i\in\beta(C)$.

Finally, suppose that $v_j\in C'$ for some $C'\in V(T)$. Then $C'$ is an $a$-cell for some $a\ge j$, and since

$\iota(v_j)\cap C'\neq\emptyset$ and $\iota(v_j)\subset C$, we conclude that $C'\subseteq C$, and consequently $C'\preceq C$.

Moreover, any cell $C''$ such that $C'\preceq C''\preceq C$ (and thus $C'\subseteq C''\subseteq C$) is an $a'$-cell

for some $a'\ge j$ and $\iota(v_j)\cap C''\supseteq\iota(v_j)\cap C'\neq\emptyset$, implying $v_j\in\beta(C'')$.

It follows that $\{C':v_j\in\beta(C')\}$ induces a connected subtree of $T$.

Finally, let us bound the width of the decomposition $(T,\beta)$. Let $C$ be a non-empty cell and let $a$ be maximum such that $C$

is an $a$-cell. Then $C$ is an open box with sides of lengths $\ell_{a,1}$, \ldots, $\ell_{a,d}$. Consider $j\in\{1,\ldots,d\}$:

\begin{itemize}

\item If $a=1$, then $\ell_{a,j}=ksd |\omega(v_a)[j]|$.

\item If $a>1$ and $\ell_{a,j}=\ell_{a-1,j}$, then $b_{a,j}=1$, implying $\ell_{a,j}=\ell_{a-1,j}<2ksd|\omega(v_a)[j]|$.

\item If $a>1$ and $\ell_{a,j}>\ell_{a-1,j}$, then $\ell_{a-1,j}\ge2ksd|\omega(v_a)[j]|$ and